23-04-22

Torsional Resistance of Standard Steel Shapes by

Carmen Chun 1 of 18

Torsional Resistance of Standard Steel Shapes

Background

Theory

Design Approaches

Worked Example

23-04-22

Torsional Resistance of Standard Steel Shapes by

Carmen Chun 2 of 18

Background

Limited guidance in the design for torsion in steel structures

CSA S16 has half a page dedicated to torsion

Torsion commonly a secondary effect to bending, shear, etc.

Situations where torsion plays a significant role in the design

Analysis for torsion covered in textbooks

Design for torsion addressed by publications by CISC and AISC

23-04-22

Torsional Resistance of Standard Steel Shapes by

Carmen Chun 3 of 18

What is torsion?

The question should be how is torsion transferred?

Torsion is carried as shear stresses in a cross-section

Cross-section rotates through an angle

23-04-22

Torsional Resistance of Standard Steel Shapes by

Carmen Chun 4 of 18

Shear Centre

The shear centre is the location in a cross-section where no torsion occurs

Forces acting through the shear centre will not cause torsional stresses

Shear centre does not have to be in the same location as the centroid

Need to determine the shear centre to evaluate the torsional stress of a section

23-04-22

Torsional Resistance of Standard Steel Shapes by

Carmen Chun 5 of 18

Pure Torsion (St. Venant Torsion)

Pure torsional shear stresses are always present in torsion

No out-of-plane warping

Torsional shear stress is proportional to the radial distance from the centre of twist

Resistance a function of: Shear modulus (G) Torsional constant of cross-section (J) First derivative of the angle of twist with

respect to the z-axis, i.e. change in angle of rotation per unit length (θ’)

T = GJθ’

23-04-22

Torsional Resistance of Standard Steel Shapes by

Carmen Chun 6 of 18

Torsional Constant (J)

J is the torsional constant of cross-section with units of length to the 4th power, i.e. mm4

For open sections:

For closed sections:

Fillets are typically ignored in sections such as single angles and structural tees

ds

t

Ao

23-04-22

Torsional Resistance of Standard Steel Shapes by

Carmen Chun 7 of 18

Warping Torsion

When the cross-section is prevented or restrained from warp freely, longitudinal bending results

This is typical in open sections, where the cross-section no longer remains plane after twisting

Axial forces are induced in the section

Resistance is a function of: Modulus of elasticity of steel (E) Warping constant for the cross-section

(Cw) Third derivative of the angle of twist with

respect to the z-axis (θ’”)

Tw = ECwθ”’

23-04-22

Torsional Resistance of Standard Steel Shapes by

Carmen Chun 8 of 18

Warping Torsional Constant (Cw)

In HSS, warping deformations are small and is generally taken as zero

Warping constant is calculated differently for various shapes.

CSA S-16 includes the values

For example, Cw for a W-shape:

Cw = Ifh2/2

23-04-22

Torsional Resistance of Standard Steel Shapes by

Carmen Chun 9 of 18

Torsional Analysis

Pure torsional shear stresses

Shear stresses due to warping

Normal stresses due to warping

Bending stresses from plane bending

Shear stresses from plane bending

Axial stress from axial load

Combine all the above stresses, but pay attention to direction!

23-04-22

Torsional Resistance of Standard Steel Shapes by

Carmen Chun 10 of 18

Design Approaches

Used closed sections

Use diagonal bracing

Make rigid end connections

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Torsional Resistance of Standard Steel Shapes by

Carmen Chun 11 of 18

Examples of Torsional Loading

Spandrel beams

Beam framing into a girder on one side only

Unequal reactions on either side of a girder

Crane runway girders

Situations where loading or reaction acts eccentrically to the shear centre

23-04-22

Torsional Resistance of Standard Steel Shapes by

Carmen Chun 12 of 18

Worked Example

MC 460 x 63.5

Span, L = 3658 mm (12 ft)

wf = 52.5 kN/m

Fixed-fixed boundary conditions

Load acts through centroid of channel

Resolve to torsional moment and load applied through the shear centre

23-04-22

Torsional Resistance of Standard Steel Shapes by

Carmen Chun 13 of 18

Worked Example (cont’d)

Torsional properties: J = 513 x 103 mm4

a = 1072 mm Cw = 227 x 103 mm4

Wno = 14.2 x 103 mm2

Wn2 = 6.7 x 103 mm2

Sw1 = 7.24 x 106 mm4

Sw2 = 5.62 x 106 mm4

Sw3 = 2.81 x 106 mm4

e0 =24.6 mm

Qf = 322.8 x 103 mm3

Qw = 621.1 x 103 mm3

Flexural properties: Ix = 231 x 106 mm4

Sx = 1010 x 103 mm3

tf = 15.9 mm

tw = 11.4 mm

23-04-22

Torsional Resistance of Standard Steel Shapes by

Carmen Chun 14 of 18

Worked Example (cont’d)

Calculate bending stresses:

At support:

Mf = wfL2/12 = 58.5 kNm

Vf = wfL/2 = 96 kN

σb = Mf/Sx = 58 MPa

τbf = Vf Qf/Ixtf = 8.4 MPa

τbw = Vf Qw/Ixtw = 22.6 MPa

At midspan:

Mf = 27.8 kNm

σb = 29 MPa

At z/l = 0.2:

Vf = 57.8 kN

τbf = 5.1 MPa

τbw = 13.7 MPa

23-04-22

Torsional Resistance of Standard Steel Shapes by

Carmen Chun 15 of 18

Worked Example (cont’d)

Calculate torsional stresses:

t = wfe = 2.5 kNm per m

L/a = 3.40z/L 0 0.2 0.5

θ 0 +0.07

0.15 taL/2GJ

θ' 0 +0.14

0 GJ/t * 2/L

θ" -1.0 0 -0.20 GJ/t* 2a/L

θ‘” +0.46

-0.46 0 GJ/t* 2a2/L

23-04-22

Torsional Resistance of Standard Steel Shapes by

Carmen Chun 16 of 18

Worked Example (cont’d)

Shear stresses due to pure torsion: At support and midspan, 0. At z/L = 0.2

14.1 MPa in the web 20 MPa in the flange

Stresses due to warping:

z/L 0 0.2 0.5

τw1 9 MPa 4.1 MPa

0

τw2 7 MPa 3.2 MPa

0

τw3 4.8 MPa

2.2 MPa

0

σwo 139MPa

0 -60 MPa

σw2 65 MPa 0 -28 MPa

23-04-22

Torsional Resistance of Standard Steel Shapes by

Carmen Chun 17 of 18

Worked Example (cont’d)

Maximum normal stress occurs at support at Point 2 in the flange

Maximum shear stress occurs at z/L = 0.2 at Point 3 in the web

Maximum rotation:

θ = +0.15 taL/2GJ

θ = 0.012 rad (0.69 deg)

23-04-22

Torsional Resistance of Standard Steel Shapes by

Carmen Chun 18 of 18

Thank you.

Thank you for your attention

Questions?